Smoothed-particle hydrodynamics

Smoothed-particle hydrodynamics (SPH) is a computational method used for simulating fluid flows. It has been used in many fields of research, including astrophysics, ballistics, volcanology, and oceanography. It is a mesh-free Lagrangian method (where the coordinates move with the fluid), and the resolution of the method can easily be adjusted with respect to variables such as the density.

Contents

Method

The smoothed-particle hydrodynamics (SPH) method works by dividing the fluid into a set of discrete elements, referred to as particles. These particles have a spatial distance (known as the "smoothing length", typically represented in equations by h ), over which their properties are "smoothed" by a kernel function. This means that the physical quantity of any particle can be obtained by summing the relevant properties of all the particles which lie within the range of the kernel. For example, using Monaghan's popular cubic spline kernel the temperature at position \mathbf{r} depends on the temperatures of all the particles within a radial distance 2h of \mathbf{r}.

The contributions of each particle to a property are weighted according to their distance from the particle of interest, and their density. Mathematically, this is governed by the kernel function (symbol W ). Kernel functions commonly used include the Gaussian function and the cubic spline. The latter function is exactly zero for particles further away than two smoothing lengths (unlike the Gaussian, where there is a small contribution at any finite distance away). This has the advantage of saving computational effort by not including the relatively minor contributions from distant particles.

The equation for any quantity A at any point \mathbf{r} is given by the equation

A(\mathbf{r}) = \sum_j m_j \frac{A_j}{\rho_j} W(| \mathbf{r}-\mathbf{r}_{j} |,h),

where m_j is the mass of particle j , A_j is the value of the quantity A for particle j , \rho_j is the density associated with particle j , \mathbf{r} denotes position and W is the kernel function mentioned above. For example, the density of particle i ( \rho_i ) can be expressed as:

\rho_i = \rho(\mathbf{r}_i) = \sum_j m_j \frac{\rho_j}{\rho_j} W(| \mathbf{r}_i-\mathbf{r}_j |,h) = \sum_j m_j W(\mathbf{r}_i-\mathbf{r}_j,h),

where the summation over j includes all particles in the simulation.

Similarly, the spatial derivative of a quantity can be obtained by using integration by parts to shift the del ( \nabla ) operator from the physical quantity to the kernel function,

\nabla A(\mathbf{r}) = \sum_j m_j \frac{A_j}{\rho_j} \nabla W(| \mathbf{r}-\mathbf{r}_j |,h).

Although the size of the smoothing length can be fixed in both space and time, this does not take advantage of the full power of SPH. By assigning each particle its own smoothing length and allowing it to vary with time, the resolution of a simulation can be made to automatically adapt itself depending on local conditions. For example, in a very dense region where many particles are close together the smoothing length can be made relatively short, yielding high spatial resolution. Conversely, in low-density regions where individual particles are far apart and the resolution is low, the smoothing length can be increased, optimising the computation for the regions of interest. Combined with an equation of state and an integrator, SPH can simulate hydrodynamic flows efficiently. However, the traditional artificial viscosity formulation used in SPH tends to smear out shocks and contact discontinuities to a much greater extent than state-of-the-art grid-based schemes.

The Lagrangian-based adaptivity of SPH is analogous to the adaptivity present in grid-based adaptive mesh refinement codes, though in the latter case one can refine the mesh spacing according to any criterion selected by the user. Because it is Lagrangian in nature, SPH is limited to refining based on density alone.

Often in astrophysics, one wishes to model self-gravity in addition to pure hydrodynamics. The particle-based nature of SPH makes it ideal to combine with a particle-based gravity solver, for instance tree gravity,[1] particle mesh, or particle-particle particle-mesh.

Uses in astrophysics

The adaptive resolution of smoothed-particle hydrodynamics, combined with its ability to simulate phenomena covering many orders of magnitude, make it ideal for computations in theoretical astrophysics.

Simulations of galaxy formation, star formation, stellar collisions, supernovae and meteor impacts are some of the wide variety of astrophysical and cosmological uses of this method.

SPH is used to model hydrodynamic flows, including possible effects of gravity. Incorporating other astrophysical processes which may be important, such as radiative transfer and magnetic fields is an active area of research in the astronomical community, and has had some limited success. [1]

Uses in fluid simulation

Smoothed-particle hydrodynamics is being increasingly used to model fluid motion as well. This is due to several benefits over traditional grid-based techniques. First, SPH guarantees conservation of mass without extra computation since the particles themselves represent mass. Second, SPH computes pressure from weighted contributions of neighboring particles rather than by solving linear systems of equations. Finally, unlike grid-base technique which must track fluid boundaries, SPH creates a free surface for two-phase interacting fluids directly since the particles represent the denser fluid (usually water) and empty space represents the lighter fluid (usually air). For these reasons it is possible to simulate fluid motion using SPH in real time. However, both grid-based and SPH techniques still require the generation of renderable free surface geometry using a polygonization technique such as metaballs and marching cubes, point splatting, or "carpet" visualization. For gas dynamics it is more appropriate to use the kernel function itself to produce a rendering of gas column density (e.g. as done in the SPLASH visualisation package).

One drawback over grid-based techniques is the need for large numbers of particles to produce simulations of equivalent resolution. In the typical implementation of both uniform grids and SPH particle techniques, many voxels or particles will be used to fill water volumes which are never rendered. However, accuracy can be significantly higher with sophisticated grid-based techniques, especially those coupled with particle methods (such as particle level sets), since it is easier to enforce the incompressibility condition in these systems. SPH for fluid simulation is being used increasingly in real-time animation and games where accuracy is not as critical as interactivity.

Uses in solid mechanics

William G. Hoover has used SPH to study impact fracture in solids. Hoover and others use the acronym SPAM (smooth-particle applied mechanics) to refer to the numerical method. The application of smoothed-particle methods to solid mechanics remains a relatively unexplored field.

References

  1. ^ "The Parallel k-D Tree Gravity Code"; "PKDGRAV (Parallel K-D tree GRAVity code" use a kd-tree gravity simulation.

External links

Software